• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.727-740
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• 2008

A ring R is called IFP, due to Bell, if ab=0 implies aRb=0 for $a,b{\in}R$. Huh et al. showed that the IFP condition need not be preserved by polynomial ring extensions. But it is shown that ${\sum}^n_{i=0}$ $E_{ai}E$ is a nonzero nilpotent ideal of E whenever R is an IFP ring and $0{\neq}f{\in}F$ is nilpotent, where E is a polynomial ring over R, F is a polynomial ring over E, and $a_i^{'s}$ are the coefficients of f. we shall use the term near IFP to denote such a ring as having place near at the IFPness. In the present note the structures of IFP rings and near-IFP rings are observed, extending the classes of them. IFP rings are NI (i.e., nilpotent elements form an ideal). It is shown that the near-IFPness and the NIness are distinct each other, and the relations among them and related conditions are examined.

• East Asian mathematical journal
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• v.25 no.4
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• pp.441-447
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• 2009

Jat and Choudhari defined a near-ring R with left bipotent or right bipotent condition in 1979. Also, we can dene a near-ring R as subcommutative if aR = Ra for all a in R. From these above two concepts it is natural to investigate the near-ring R with the properties aR = $Ra^2$ (resp. $a^2R$ = Ra) for each a in R. We will say that such is a near-ring with (1,2)-potent condition (resp. a near-ring with (2,1)-potent condition). Thus, we can extend a general concept of a near-ring R with (m,n)-potent condition, that is, $a^mR\;=\;Ra^n$ for each a in R, where m, n are positive integers. We will derive properties of near-ring with (1,n) and (n,1)-potent conditions where n is a positive integer, any homomorphic image of (m,n)-potent near-ring is also (m,n)-potent, and we will obtain some characterization of regular near-rings with (m,n)-potent conditions.

• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.663-683
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• 2015

In this note we study the structures of power-serieswise Armendariz rings and IFP rings when they are skewed by ring endomor-phisms (or automorphisms). We call such rings skew power-serieswise Armendariz rings and skew IFP rings, respectively. We also investigate relationships among them and construct necessary examples in the process. The results argued in this note can be extended to the ordinary ring theoretic properties of power-serieswise Armendariz rings, IFP rings, and near-related rings.

• Journal of applied mathematics & informatics
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• v.33 no.3_4
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• pp.469-474
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• 2015

In this paper, we will consider the notions of (m, n)-potent conditions in near-rings, in particular, a near-ring R with left bipotent or right bipotent condition. We will derive some properties of near-rings with (1, n) and (n, 1)-potent conditions where n is a positive integer, and then some properties of near-rings with (m, n)-potent conditions. Also, we may discuss the behavior of R-subgroups in (1, n)-potent or (n, 1)-potent near-rings..